% 1

% see slides W11

% 2

expr --> "0".
expr --> "1".
expr --> "x1"; "x2"; "x3".
expr --> "-",
	 "(",
	 expr,
	 ")".
expr --> "(",
	 expr,
	 ".",
	 expr,
	 ")".
expr --> "(",
	 expr,
	 "+",
	 expr,
	 ")".

:- phrase(expr,`(1+1)`).

expr(0) --> "0".
expr(1) --> "1".
expr(var) --> "x1"; "x2"; "x3".
expr(comp(E)) --> "-",
	 "(",
	 expr(E),
	 ")".
expr(times(E1,E2)) --> "(",
	 expr(E1),
	 ".",
	 expr(E2),
	 ")".
expr(plus(E1,E2)) --> "(",
	 expr(E1),
	 "+",
	 expr(E2),
	 ")".

:- phrase(expr(_),`(1+1)`).

expr_improved(0) --> "0".
expr_improved(1) --> "1".
expr_improved(var) --> "x1"; "x2"; "x3".
expr_improved(comp(E)) -->
    "-",
    lbracket,
    expr_improved(E),
    rbracket.
expr_improved(times(E1,E2)) --> "(",
    expr_improved(E1),
    blanks,
    ".",
    blanks,
    expr_improved(E2),
    ")".
expr_improved(plus(E1,E2)) --> "(",
    expr_improved(E1),
    blanks,
    "+",
    blanks,
    expr_improved(E2),
    ")".

lbracket --> "("; "".

rbracket --> ")"; "".

blanks --> " "; "".

:- phrase(expr_improved(_),`(-1 + -1)`).

% 4

:- use_module(library(clpfd)).

plan([A,B,C,D,E,F]) :-
    A #< B,
    A #< C,
    A #< D,
    B #\= C,
    C #\= D,
    B #< F,
    C #< E,
    D #< E,
    E #< F,
    [A,B,C,D,E,F] ins 1..6,
    label([A,B,C,D,E,F]).

% 5

profile(true,0).
profile((A,B),N) :-
    profile(A,N1),
    profile(B,N2),
    N is max(N1,N2).
profile(A,N) :-
    dif(A,(_,_)),
    dif(A,true),
    clause(A,B),
    profile(B,N0),
    N is N0 + 1.